# THEORY OF STRONG INTERACTIONS

**Academic Year 2022/2023**- Teacher:

**VINCENZO GRECO**

## Expected Learning Outcomes

The course aims to introduce the basic elements of quantum-relativistic field theory in order to provide the basis for the understanding of the modern theory of fundamental interactions. After a critical exposition of relativistic quantum mechanics, the main arguments concerning the formulation of a relativistic quantum field theory are exposed. The main objective is learning Quantum ElectroDynamics (QED), an abelian gauge theory, as a prototype of the modern description of a fundamental interaction, and then formulation and properties of non-abelian gauge theory for strong interaction: the QuantumChromoDynamics (QCD).

Upon completion of the course the student has to be able to calculate the cross section of the elemmentary processes in both QED and QCD. This will be achieved giving during the frontal lecture a very large space for explicit calculations with all students. Also several examples will be worked out with the explicit calculation of cross sections and decays in the classroom during the course itself.

## Course Structure

The Course is structured by meant of frontal lectures for both the theoretical part ( 5 CFU - 35 hours) and for the exercises (1 CFU - 15 hours). There will be however also some exercise during the class held as practical tests to verify the understanding of some main part of the course. Should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

**Introduction** - Dimensional analysis, senergy and time scale for process under the action of the foru fundamental forces. Primer on Lorentz group. Evoltuion from relativstic quantum mechanics to quantum field theroy. Lagrangian theory for classical fields and Eulero-Lagrange equation.Hamiltonian formulation. Symmetries: internal e space-time, discrete e continous, global an local gauge. Noether Theorem. Energy-Momentum Tensor. Example for a scalr, spinorial and electromagnetic field.

**Quantization of Fields **- Canonical quantization for scalar, spinors and elettromagnetic fields. Spin-Statistic theorem. Interacting fields. Propagators for bosons and fermions. Pertubation theory in quantum field theory. Normal and temporal products and Wick Theorem. Definitions of S , T and M matrice and their relation to cross sections and particle decay. Feymann digramas and primer on radiative corrections. Mandelstam variables and their use for the cross section: s-channel, t-channel ed u-channel. Crossing symmetry. Link of non-relativistic Bron approximation and leading order diagram in quantum field theory.

**Quantum ElecttroDynamics– QED** - Gauge field, minimal coupling and QED formualtion. Feynmann rules for QED. Coulomb Scattering for elettrons e positrons and non-relativistic reduction to Rutherford cross section; Electron scattering: Moeller cross section; Scattering elettron-positron: Babbha scattering; Scattering Compton and its Ultra-relativistic limit (production of energetic photons by laser).

**Quantum Chromodynamics- QCD** - Introduction to Stron Interactions and QCD lagragian; some proof for quarks and their flavor and color. on abelian gauge interaction in SU(2) e extension to SU(3). Non abelian bosonic tensor. Feynmann diaggrams in QED e QCD. Asymptotic freedom and Confinement. Chiral symmetry and transition from adronic matter to a plasma of quarks and gluons. Nuclear Interaction as meson exchange and Quantum HadroDinamics (QHD). Noon-perturbative phenomena of strong interactions. Deep-inelastic collisions and form factors and parton model. Derivation of Rosenbluth formula for hadronic scatterings: electric and magnetic form factors. Bjorken scaling and parton distribution functions. QCD diagrams at leading order (qq→qq, gg→gg, qg→qg …) and cross section evaluation for elementary processes at ultra-relativistic energies. Calcualtion of hadronic spectra proton-proton, proton-nucleus e nucleus-nucleus collisions at relativistic energy.

## Textbook Information

1) F. Mandl and G. Shaw, *Quantum Field Theory*, Ed. Wiley- 1993

2) M. Maggiore, *A Modern Introduction to Quantum Field Theory*, Ed. Oxford University Press-2005

3) F.Halzen and A.D.Martin, *Quarks and leptons: an introductory course in particle physics*, Ed. Wiley 1984

4) M.E. Peskin and D.V. Schroeder, *An Introduction to Quantum Field Theory*, Ed. Westview Press- 1995

5) F. Schwabl, Advanced Quantum Mechanics, Ed. Springer.

6) G. Sterman et al., “*Handbook of Perturbative QCD*”, Review of Modern Physics 67 (1995) 158.

## Course Planning

Subjects | Text References | |

1 | Quantizzazione canonica di un campo scalare e spinoriale | M. Maggiore, A Modern Introduction to Quantum Field Theory |

2 | Espansione perturbativa della matrice di scattering | F. Mandl and G. Shaw |

3 | Diagrammi di Feynmann in QCD e QED e loro calcolo | F. Mandl and G. Shaw/M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory |

4 | Modello a partoni e fattori di forma | F.Halzen and A.D.Martin, Quarks and leptons: an introductory course in particle physics |

5 | Equazioni di Eulero in teoria dei campi | A Modern Introduction to Quantum Field Theory |

6 | Teorema di Noether per simmetrie interne e spazio-temproali | A Modern Introduction to Quantum Field Theory |

7 | Libertà asintotica e confinamento | F.Halzen and A.D.Martin, Quarks and leptons: an introductory course in particle physics |

8 | Interazione nucleare dallo scambio di mesoni | J. D. Walecka, Theoretical Nuclear and Subnuclear Physics, |

9 | Derivazione teoria di gauge non abeliana | M. Maggiore, A Modern Introduction to Quantum Field Theory, |

10 | PRroblema della causalità in teoria dei campi | M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory |

11 | Produzione di adroni in collisioni ultra-relativistiche | G. Sterman et al., “Handbook of Perturbative QCD”, |

## Learning Assessment Procedures

Oral exam with at least 3-4 questions mainly on the following topics: motivation for a quantum-relativistic field theory, classical field theory and Noether's theorem, canonical quantization, perturbative theory in field theory and Feynmann diagrams, abelian gauge theory and QED, non-abelian gauge theory and QCD, parton model, chiral symmetry. They will also be asked to show that they can write the scattering matrix of elementary processes starting from Feynmann diagrams. Verification of learning can also be carried out electronically, should the conditions require it.## Examples of frequently asked questions and / or exercises

The questions below are not an exhaustive list but are just a few examples.

The most frequently asked question concerns the calculation of the scattering matrix for both QED and QCD processes. Other frequently asked questions concern: the canonical quantization for a scalar or Dirac field, the Noether theorem, the formulation of the non-abelian gauge theory, the parton model and chiral symmetry.